اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIterated Random Functions and Slowly Varying Tails
33
0
0.0
(
0
)
تأليف
Piotr Dyszewski
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Consider a sequence of i.i.d. random Lipschitz functions ${Psi_n}_{n geq 0}$. Using this sequence we can define a Markov chain via the recursive formula $R_{n+1} = Psi_{n+1}(R_n)$. It is a well known fact that under some mild moment assumptions this Markov chain has a unique stationary distribution. We are interested in the tail behaviour of this distribution in the case when $Psi_0(t) approx A_0t+B_0$. We will show that under subexponential assumptions on the random variable $log^+(A_0vee B_0)$ the tail asymptotic in question can be described using the integrated tail function of $log^+(A_0vee B_0)$. In particular we will obtain new results for the random difference equation $R_{n+1} = A_{n+1}R_n+B_{n+1}$..
قيم البحث
اقرأ أيضاً
In this article, we consider a Branching Random Walk (BRW) on the real line where the underlying genealogical structure is given through a supercritical branching process in i.i.d. environment and satisfies Kesten-Stigum condition. The displacements
coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (depends on the expected size of the $n$-th generation given the environment) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Frech{e}t random variable. Furthermore, we derive the weak limit of the extremal processes composed of appropriately scaled positions at the $n$-th generation and show that the limit point process is a member of the randomly scaled scale-decorated Poisson point processes (SScDPPP). Hence, an analog of the predictions by Brunet and Derrida (2011) holds.
We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.s. It turns out that the idea of hypercontractivity for
minima is closely related to small ball probabilities and Gaussian correlation inequalities.
Let A be a finite subset of an abelian group (G, +). Let h $ge$ 2 be an integer. If |A| $ge$ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + $times$ $times$ $times$ + A is known, what can one say about |(h -- 1)A| and |(h + 1)A|? It
is known that |(h -- 1)A| $ge$ |hA| (h--1)/h , a consequence of Pl{u}nneckes inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h$ge$0 with the Hilbert function of a standard graded algebra. We then apply Macaulays 1927 theorem on the growth of Hilbert functions, and more specifically a recent condensed version of it. Our bound implies |(h -- 1)A| $ge$ $theta$(x, h) |hA| (h--1)/h for some factor $theta$(x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that $theta$(x, h) asymptotically tends to e $approx$ 2.718 as |A| grows and h lies in a suitable range varying with |A|.
We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $mathbb{P}[X geq t] sim a exp(
-lambda t^r)$ for some constants $a, lambda > 0$ where $r in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r in (0, 2/3)$ and $ r in (2/3, 1)$, with yet different limits in the boundary case $r = 2/3$.
Let $(xi_k,eta_k)_{kinmathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{kinmathbb{N}}$ defined by $T_k:=xi_1+cdots+xi
_{k-1}+eta_k$ for $kinmathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $jinmathbb{N}$ and $tgeq 0$, denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $leq t$. In this article we prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwells theorem and the key renewal theorem) for $N_j(t)$ under the assumption that $j=j(t)toinfty$ and $j(t)=o(t^{2/3})$ as $ttoinfty$. According to our terminology, such generations form a subset of the set of intermediate generations.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
